Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics

Abstract

We give a classification of generic coadjoint orbits for the groups of symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic surface. We also classify simple Morse functions on symplectic surfaces with respect to actions of those groups.This gives an answer to V.Arnold's problem on describing all invariants of generic isovorticed fields for the 2D ideal fluids. For this we introduce a notion of anti-derivatives on a measured Reeb graph and describe their properties.

Characterization of Steady Solutions to the 2D Euler Equation

“…iii) the volume of Γ with respect to the measure µ is equal to the volume of M : When M is closed, this result is Theorem 3.11 of [10], and, as we explain below, the general case can be reduced to the case of closed M by means of the symplectic cut operation.…”

“…Therefore, we have a one-to-one correspondence between triples pM, F, ωq (with boundary) and triples pM ,F ,ωq (without boundary) with marked minima and maxima. Since measured Reeb graphs up to isomorphism completely describe simple Morse functions on closed symplectic surfaces modulo symplectomorphisms (see Theorem 3.11 of [10]), this one-to-one correspondence extends to symplectic surfaces with boundary.…”

Classification of Casimirs in 2D Hydrodynamics

“…This methods turned out to be effective for affine groups and the Virasoro-Bott group, so one may hope to apply it to 2D diffeomorphisms and current groups as well. Finally, note that all objects in the present paper are infinitely smooth (see the case of finite smoothness in [6]). To the best of our knowledge, a complete description of Casimirs in 2D fluid dynamics has not previously appeared in the literature in a self-contained form, while various partial results could be found in [3,9,10,11,5].…”

“…While it has been known for a long time that enstrophies are first integrals of 2D incompressible fluid flows, a complete classification of generic Casimirs in 2D was obtained only recently in [6,5]. Here we revisit and develop that classification by comparing it to other known classification of coadjoint orbits for diffeomorphism groups in one dimension.…”

Morse–Darboux lemma for surfaces with boundary

“…The Morse-Darboux lemma is a particular case of Le lemme de Morse isochore, see [1], and also a particular case of Eliasson's theorem on the normal form for an integrable Hamiltonian system near a non-degenerate critical point, see [2,3]. The Morse-Darboux lemma is an important tool in topological hydrodynamics, see [4], and theory of integrable systems, see [5]. We expect that the result of the present paper will also be useful in 2D fluid dynamics.…”